**MSC:**- 54E05 Proximity structures and generalizations
- 54E15 Uniform structures and generalizations
- 54B20 Hyperspaces
- 54A10 Several topologies on one set (change of topology, comparison of topologies, lattices of topologies)
- 54D30 Compactness

hyperspace topologies. A proximal hypertopology corresponding to a

LO-proximity is a g eneralization of the well known Vietoris

topology. In case we start with an EF-proximity, the proximal

hypertopology equals the Hausdorff uniform topology corresponding

to the totally bounded uniformity and, being contained in both the

Vietoris and Hausdorff uniform topologies, serves as a bridge

between the two. Wattenberg and Beer-Himmelberg-Prickry-Van Vleck

showed that the locally finite hypertopology induced by a

metrizable space is the sup of the Hausdorff metric topologies

induced by all compatible metrics. Naimpally-Sharma showed that

this follows from the fact that a Tychonoff space is normal iff

its fine uniformity induces the locally finite hypertopology. Di

Concilio-Naimpally-Sharma showed that in a Tychonoff space the

fine uniformity induces the proximal locally finite hypertopology.

We study DELTA topologies introduced by Poppe, and their proximal

variations. We show that a short proof can be given of the

Beer-Tamaki result concerning the uniformizability of (proximal)

DELTA hypertopologies via the Attouch-Wets approach used by Beer

in dealing with the Fell topology. Finally we present a result

concerning (Proximal) DELTA-U-hypertopolgies. Several new

hypertopologies are introduced.